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One-dimensional Potts model, Lee-Yang edges, and chaos
1Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland.
Summary
The study reveals that the one-dimensional q-state Potts model, like the Ising model, exhibits renormalization transformations related to the logistic map. Its Lee-Yang zeros map onto the logistic map
Area of Science:
- Statistical Mechanics
- Complex Systems
Background:
- Renormalization transformations for the one-dimensional Ising model in a field are linked to the logistic map f(x)=4x(1-x).
- The Lee-Yang zeros of the Ising model correspond to the Julia set of this logistic map.
Purpose of the Study:
- To investigate if the one-dimensional q-state Potts model exhibits similar behavior to the Ising model.
- To explore the relationship between the q-state Potts model's couplings and the logistic map.
Main Methods:
- Defining a suitable combination of couplings for the q-state Potts model.
- Analyzing the behavior of Lee-Yang zeros in the complex plane for q not equal to 2.
- Mapping the locus of these zeros using a variable 'x' derived from the model's couplings.
Main Results:
- The one-dimensional q-state Potts model (for q>=1) also shows renormalization transformations representable by the logistic map.
- A generalized 'x' variable, dependent on model couplings, satisfies f(x)=4x(1-x).
- While Lee-Yang zeros deviate from the unit circle for q!=2, their locus maps to the logistic map's Julia set.
Conclusions:
- The connection between renormalization transformations and the logistic map extends from the Ising model to the broader q-state Potts model.
- The Julia set of the logistic map serves as a unifying geometric representation for the Lee-Yang zeros across these models.